Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

Let $$X$$ be a class of extended numerical functions on domain $$D$$ $$d$$ -dimensional Euclidean space $$\mathbb{R}^{d}$$ , $$H\subset X$$ . Given $$u\in and $$M\in we write $$u\prec_{H}M$$ if there is function $$h\in H$$ such that $$u+h\leq M$$ We consider this special preorder $$\prec_{H}$$ for pair subharmonic $$u$$ $$M$$ in cases when $$H$$ the all harmonic or convex cone $$h\not\equiv-\infty$$ Main results are dual equivalent forms terms balayage processes Riesz measures Jensen Arens–Singer (representing) measures, potentials these test generated by complements $$D\backslash S$$ non-empty precompact subsets $$S\Subset D$$ Applications to holomorphic $$f$$ $$D\subset\mathbb{C}^{n}$$ relate distribution zero sets under upper restrictions $$|f|\leq\exp If $$D\subset\mathbb{C}$$ finitely connected with exterior simply two different points boundary then our conditions zeros $$f\neq 0$$ both necessary sufficient.